(a) The generalised Boltzmann distribution $P(\mathbf{p})$ is given by

$$P(\mathbf{p})=\frac{e^{-\beta\left(E_{\mathbf{p}} n_{\mathbf{p}}-\mu n_{\mathbf{p}}\right)}}{\mathcal{Z}_{\mathbf{p}}}$$

where $\beta=\left(k_{B} T\right)^{-1}, \mu$ is the chemical potential,

$$\mathcal{Z}_{\mathbf{p}}=\sum_{n_{\mathbf{p}}} e^{-\beta\left(E_{\mathbf{p}} n_{\mathbf{p}}-\mu n_{\mathbf{p}}\right)}, \quad E_{\mathbf{p}}=\sqrt{m^{2} c^{4}+p^{2} c^{2}} \quad \text { and } \quad p=|\mathbf{p}|$$

Find the average particle number $\langle N(\mathbf{p})\rangle$ with momentum $\mathbf{p}$, assuming that all particles have rest mass $m$ and are either

(i) bosons, or

(ii) fermions .

(b) The photon total number density $n_{\gamma}$ is given by

$$n_{\gamma}=\frac{2 \zeta(3)}{\pi^{2} \hbar^{3} c^{3}}\left(k_{B} T\right)^{3}$$

where $\zeta(3) \approx 1.2$. Consider now the fractional ionisation of hydrogen

$$X_{e}=\frac{n_{e}}{n_{e}+n_{H}}$$

In our universe $n_{e}+n_{H}=n_{p}+n_{H} \approx \eta n_{\gamma}$, where $\eta \sim 10^{-9}$ is the baryon-to-photon number density. Find an expression for the ratio

$$\frac{1-X_{e}}{X_{e}^{2}}$$

in terms of $\eta,\left(k_{B} T\right)$, the electron mass $m_{e}$, the speed of light $c$ and the ionisation energy of hydrogen $I \approx 13.6 \mathrm{eV}$.

One might expect neutral hydrogen to form at a temperature $k_{B} T \sim I$, but instead in our universe it happens at the much lower temperature $k_{B} T \approx 0.3 \mathrm{eV}$. Briefly explain why this happens.

[You may use without proof the Saha equation

$$\frac{n_{H}}{n_{e}^{2}}=\left(\frac{2 \pi \hbar^{2}}{m_{e} k_{B} T}\right)^{3 / 2} e^{\beta I}$$

for chemical equilibrium in the reaction $\left.e^{-}+p^{+} \leftrightarrow H+\gamma .\right]$